How to subtract complex numbers

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SAT Math › How to subtract complex numbers

Questions 1 - 3

From $7 + i \sqrt {7 }$ , subtract its complex conjugate.

$2i \sqrt {7 }$

$-2i \sqrt {7 }$

Explanation

The complex conjugate of a complex number is . Therefore, the complex conjugate of $7 + i \sqrt {7 }$ is $7 - i \sqrt {7 }$ ; subtract the latter from the former by subtracting real parts and subtracting imaginary parts, as follows:

$(7 + i \sqrt {7 })- (7 - i \sqrt {7 })$

$= 7- 7 + i \sqrt {7 }- (- i \sqrt {7 })$

$= i \sqrt {7 }+ i \sqrt {7 }$

$=2i \sqrt {7 }$

From $17 - i \sqrt{17}$ , subtract its complex conjugate.

$- 2 i \sqrt{17}$

$2 i \sqrt{17}$

Explanation

The complex conjugate of a complex number is . Therefore, the complex conjugate of $17 - i \sqrt{17}$ is $17 + i \sqrt{17}$ ; subtract the latter from the former by subtracting real parts and subtracting imaginary parts, as follows:

$(17 - i \sqrt{17})- (17 + i \sqrt{17})$

$= 17 - i \sqrt{17} - 17 - i \sqrt{17}$

$= 17 - 17 - i \sqrt{17} - i \sqrt{17}$

$= - 2 i \sqrt{17}$

From $\frac{3}{2} + \frac{1}{4} i$ , subtract its complex conjugate. What is the difference ?

$\frac{1}{2} i$

$-\frac{1}{2} i$

$\frac{7}{4} + \frac{7}{4} i$

Explanation

The complex conjugate of a complex number is , so $\frac{3}{2} + \frac{1}{4} i$ has $\frac{3}{2} - \frac{1}{4} i$ as its complex conjugate. Subtract the latter from the former:

$\left (\frac{3}{2} + \frac{1}{4} i \right ) - \left ( \frac{3}{2} - \frac{1}{4} i \right )$

$= \frac{3}{2} + \frac{1}{4} i - \frac{3}{2}+ \frac{1}{4} i$

$= \frac{3}{2}- \frac{3}{2} + \frac{1}{4} i + \frac{1}{4} i$

$= \frac{1}{2} i$

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